reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem Th54:
  f^ is total iff f"{0} = {} & f is total
proof
  thus f^ is total implies f"{0} = {} & f is total
  proof
    assume f^ is total;
    then
A1: dom (f^) = C;
    f"{0} c= C;
    then f"{0} c= dom f \ f"{0} by A1,Def2;
    hence f"{0} = {} by XBOOLE_1:38;
    then C = dom f \ {} by A1,Def2;
    hence dom f = C;
  end;
  assume that
A2: f"{0} = {} and
A3: f is total;
  thus dom (f^) = dom f \ f"{0} by Def2
    .= C by A2,A3;
end;
